Application of Monte Carlo Methods in Finance

ALLEGHENY COLLEGE DEPT. OF PHYSICS

APPLICATION OF MONTE CARLO

METHODS IN FINANCE

Joshua R. Lawrence

April 21, 2015

ABSTRACT

There is often a misconception that nothing useful can be found from “random” processes.

Physicists realized this was not the case upon close observation of Brownian motion. This

realization not only had an impact in biology, but also in many other fields such as finance.

Stock prices vary similar to how a Brownian particle fluctuates in position. These connections

between stock price fluctuations and Brownian motion can be leveraged to price options which

are contracts to trade stocks. I explore the details of these connections with the aim to eventually

develop a more realistic, and hopefully more accurate, option pricing model. I also review the

current major option pricing model, the Black-Scholes model, which can be compared against

the developed, more realistic models.

ACKNOWLEDGEMENTS

First and foremost, I would like to thank Dr. Shafiq Rahman. Not only has he helped

guide me on this project, but also along my undergraduate career. I owe a great deal of my recent

success to him and his advice. I would also like to thank my second reader, Dr. George Paily.

Both Dr. Rahman and Dr. Paily provided me with key guidance and advice for my research path.

I would like to express sincere gratitude to all of my friends (especially Shannon Petersen) that I

have made in the Allegheny College Physics Department. My friends supported me at every

setback along my research experience, which made it easier to continually move forward. I

would also like to convey gratitude to all of the Department faculty members for always pushing

my mind to its full potential. The faculty has not only taught me material, but truly changed the

way I think in my everyday life. Most importantly, I would like to thank my father for stepping

up after my mother's passing and rebuilding a support system for me at home. That support

system helped me alleviate any stress developed while away from home. Though my mother was

not able to directly aid me along this adventure, I am extremely thankful for the foundation she

laid out for me, which has pushed me along my research path. Finally, thank you to all who

either directly or indirectly helped me along this venture.

TABLE OF CONTENTS

1. Introduction ................................................................................................................................. 1

2. Derivatives .................................................................................................................................. 3

2-1 Forward vs. Future Contracts ................................................................................................ 3

2-2 Options .................................................................................................................................. 4

2-3 Put vs. Call ............................................................................................................................ 4

2-4 European vs. American Style Options .................................................................................. 5

3. Stochastic Processes.................................................................................................................... 7

3-1 Brownian Motion .................................................................................................................. 8

3-2 Stock Price Movements ........................................................................................................ 8

4. Numerical Solutions.................................................................................................................. 11

4-1 Random Walk ..................................................................................................................... 13

4-2 Relationship with Time ....................................................................................................... 15

4-3 Stock Price Evolution ......................................................................................................... 18

5. Analytical Solutions .................................................................................................................. 19

5-1 Black-Scholes Equation ...................................................................................................... 20

5-2 Black-Scholes Formula ....................................................................................................... 22

5-3 Understanding the Black-Scholes Solution ........................................................................ 24

6. Future Work .............................................................................................................................. 27

7. Appendix ................................................................................................................................... 29

A. Mathematica ......................................................................................................................... 29

B. Central Limit Theorem ......................................................................................................... 31

C. Ito’s Lemma .......................................................................................................................... 32

D. Works Cited .......................................................................................................................... 34

E. Bibliography ......................................................................................................................... 35

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1. INTRODUCTION

In 1827, Robert Brown observed the motion of pollen grains suspended in water [1]. He

discovered that both nonliving and living particles would follow an erratic motion which was not

related to the flow of the liquid. The irregular behavior of the suspended particles was later

labeled Brownian motion. Einstein and Smoluchowski realized that the cause of the strange

motion was from the random collisions between the molecules of the liquid with the suspended

particles [2]. In 1905, Einstein derived the partial differential diffusion equation for Brownian

motion along with the relationship between the mean squared displacement and time. However,

Einstein’s work would not have been possible without the work of Louis Bachelier, commonly

referred to as the father of financial mathematics [3]. In 1900, Bachelier identified the

distribution function for Wiener processes (the underlying process of Brownian motion) in his

thesis, “Theory of Speculation” [4].

Bachelier’s contributions begs the question, “What are the similarities between stock

price fluctuations and a particle undergoing Brownian motion?” Consider the pollen grain

suspended in the flowing liquid. The grain is steered by the flow of the liquid. Similarly, the

stock price is driven by some growth rate depending on the success or failure of the stock’s

company. The grain will also experience some degree of random movement from the liquid

domain. The stock price will also randomly fluctuate due to their complex environment. This

complex environment results from many direct and indirect impacts such as trade rumors,

climate, psychological factors, war, and many other contributors to the complex economy of

today’s society. This paper seeks to explore these connections between the Brownian particle and

stock price in order to create more realistic, and hopefully more accurate, pricing models.

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2. DERIVATIVES

There are three main categories of financial instruments: debt, equity, and derivatives.

Debts provide investors with repayment of principal plus interest at some predetermined

future date.

Equities essentially provide shareholders part ownership of a company. The amount that

the shareholders own is called the owners’ interest, which is equal to the company’s

assets minus its liabilities. Compared to investors, shareholders have elevated risk that is

based on the company’s performance.

Derivatives are a form of contract that “derive” their value from an underlying entity.

Derivatives provide a way to hedge, or reduce risk. The most common derivatives are

forward and future contracts.

2-1 FORWARD VS. FUTURE CONTRACTS

There are several types of derivatives. Two basic examples of derivatives are forward and

future contracts. We as consumers are typically accustomed to a spot market in which trading

occurs immediately upon agreement. Forward and future contracts occur in a futures market in

which trading occurs later in the future after the agreement. While both are contracts to buy or

sell assets at a predefined date (maturity or expiration date) and price (strike price), there are a

few fundamental differences between the two. The most general difference separating forward

and future contracts is the formality. Future contracts are standardized and are traded over a

public exchange whereas forward contracts are negotiated privately.

Since future contracts are handled publicly, there is a lower counterparty risk. This means

that there is a smaller chance that one of the sides cannot follow through with the agreement. The

exchange which handles future contracts acts as the counterparty to both sides ensuring that they

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can follow through with the contract. Stock trading takes place in the form of future contracts

rather than forward contracts. The buyer of a futures contract, or long position, agrees to buy the

underlying entity on the maturity date at the strike price whereas the seller of a futures contract,

or short position, agrees to sell the underlying entity at that price on the agreed upon date.

However, future contracts rarely mature at the maturity date. Future contracts are frequently

traded or closed before the maturity date. To close the contract, the buyer (or long position) must

sell (or short) the contract. Alternatively, the seller (or short position) must buy (or long) the

contract.

2-2 OPTIONS

Options can be thought of as a form of future contract in which the owner of the contract

has the right rather than the obligation to trade an underlying asset at a strike price on the

maturity date. This right to trade costs a premium. If the owner decides to trade, this is called

exercising the option. There are two types of options: puts and calls.

2-3 PUT VS. CALL

A call is the right to buy a stock on or before a predefined maturity date at a predefined

strike price. A put is similar to a call except that it is the right to sell a stock rather than the right

to buy [5]. Both calls and puts cost a premium for that right. Calls are purchased in hopes of the

stock value increasing and puts are purchased in hopes of the stock value decreasing. For

example, consider a stock that is currently priced at $100 and you buy a call option at a premium

of $10 for a strike price of $110 to expire in a month. If the stock value goes up to $121 before

the maturity date, then you may exercise the call to buy the stock for $110 and sell it for $121

profiting $1 due to the premium. Likewise, if a stock is currently priced at $100 and you buy a

put option at a premium of $10 for a strike price of $90 to expire in a month. If the stock value

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goes down to $79 before the maturity date then you may buy the stock for $79 and exercise the

put to sell the stock for $90 profiting $1. If the value of the stock does not change to the desired

amount by the maturity date then there is only a loss of $10 from the premium. While some types

of options may be exercised before the maturity date, other options can only be exercised on the

maturity date.

2-4 EUROPEAN VS. AMERICAN STYLE OPTIONS

While European options can only be exercised at the maturity date, American options may be

exercised at any time before the maturity date. The increased flexibility of the American option

results in more difficult decision making compared to the European option. If the owner of the

option decides to exercise before maturity, he/she forgoes the opportunity to profit a larger

amount in the remaining time until expiration. Of course, he/she may also be avoiding a potential

loss. As a result, American options tend to be more complex to handle and more difficult to

model. Since they are more difficult to model, numerical simulations might give us good insight

about how to appropriately price these options.

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3. STOCHASTIC PROCESSES

Given that physics studies nature at a fundamental level, it has strong overlaps with a

variety of fields from biology to economics. This project seeks to look further into a connection

between physics and finance typically called ‘econophysics.’ The stock market has numerous

ties to physics, one of which is the overall market’s relation to multifractals in chaos theory.

Similar to Erin Brown’s Senior Comprehensive Project, in which the “sudden flips” of the

second pendulum of a double pendulum might be predicted through simulation, it is possible that

computer simulation could predict sudden and significant market changes.

The flow of the price of a specific stock or derivative has a physical analog as well. In

biophysics, when a particle is suspended in a fluid, it is considered to be undergoing Brownian

motion if the random collisions between the particle and the fluid’s molecules influence the

particle’s motion. The influence from the collisions causes the particle’s motion to be a

stochastic process. A stochastic process incorporates random fluctuations which cause the

evolution of the system over time to be probabilistic, i.e. the trajectory is not exact, but lies

within a band. While stochastic processes were originally discovered from the behavior of

physical systems, connections to non-physical systems, such as stock prices, were made very

soon thereafter [6]. Stock prices fluctuate from the random interactions with their environment.

These interactions may be direct such as the effects weather can have on grain stocks. The

interactions may also be from a summation of many indirect variables resulting in random

effects. As a result, the stock price fluctuation can be considered a Wiener process similar to

Brownian motion. A Wiener process is simply a continuous time stochastic process. The relation

between a particle undergoing Brownian motion and stock price movements is explained further

in the next section.

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3-1 BROWNIAN MOTION

Newton’s laws of motion can be applied to understand balance of forces. In biophysics,

balance of forces can be applied with a Langevin equation to understand Brownian motion. The

Langevin equation describing a particle undergoing Brownian motion is as follows:

𝐹𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 = 𝐹𝑑𝑟𝑎𝑔 + 𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔 + 𝐹𝑠𝑡𝑜𝑐ℎ𝑎𝑠𝑡𝑖𝑐

𝑚𝑑2𝑥

𝑑𝑡2 = −𝛾𝑑𝑥

𝑑𝑡− 𝑘𝑥 + 𝐹𝑠𝑡𝑜𝑐ℎ𝑎𝑠𝑡𝑖𝑐 (1)

where x is the particle’s position, t is time, m is the mass of the particle, 𝛾 is the drag coefficient

from the viscosity of the fluid, k is the return factor from Hooke’s law, and 𝐹𝑠𝑡𝑜𝑐ℎ𝑎𝑠𝑡𝑖𝑐 is the

noise term that comes from the random interactions between the particle and the fluid. It should

be noted that 𝐹𝑠𝑡𝑜𝑐ℎ𝑎𝑠𝑡𝑖𝑐 has no meaning by itself and is just a noise term for the balance of forces

in the Langevin equation. Equation (1) results from three different interactions between the

particle and the fluid environment. The first comes from the fluid resisting the motion of the

particle in the form of drag friction. The second interaction is the fluid’s tendency to draw the

particle back to its original position due to weak attraction between the particle and the

molecules of the surrounding fluid. The final interaction results from the collision of the particle

with the fluid’s molecules. The collisions are random and hence characterize the process to be

stochastic.

3-2 STOCK PRICE MOVEMENTS

Observing physical systems undergoing Brownian motion marked the discovery of

stochastic processes [7]. Application to non-physical fields, such as finance, began soon

thereafter [8]. It was realized that stock prices also change stochastically with time. Since Monte-

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Carlo simulations numerically solve problems through repeated random samplings, the

“randomness” from stochastic processes can be effectively modeled using Monte-Carlo

simulations, which will be shown further in section 4.

Stock pricing incorporates the same interactions as equation (1) except that it does not

have a restoring force and so equation (1) can be rewritten as follows:

𝑑2𝑥

𝑑𝑡2= −

𝛾

𝑚

𝑑𝑥

𝑑𝑡+

1

𝑚𝐹𝑠𝑡𝑜𝑐ℎ𝑎𝑠𝑡𝑖𝑐 .

𝑑𝑉

𝑑𝑡= −

𝛾

𝑚 𝑉 +

1

𝑚𝐹𝑠𝑡𝑜𝑐ℎ𝑎𝑠𝑡𝑖𝑐

𝑑𝑉 = −𝛾

𝑚 𝑉𝑑𝑡 +

1

𝑚𝐹𝑠𝑡𝑜𝑐ℎ𝑎𝑠𝑡𝑖𝑐𝑑𝑡 (2)

where V is the particle’s velocity. Several changes can be made to equation (2) to translate it

from describing the velocity of a particle undergoing Brownian motion to stock pricing. Though

these changes are not direct, they are still vital for conceptualizing stock price movements. The

right side of equation (2) can be split into two parts, deterministic and stochastic. The

deterministic part comes from the drag force which hinders the velocity of the particle. The

stochastic part comes from the stochastic “force” which randomly fluctuates the velocity of the

particle. As a result the particle’s velocity changes due to both the deterministic and stochastic

contributions. Similarly, a stock’s price, 𝑆(𝑡), changes due to both deterministic and stochastic

contributions. While the deterministic drag force resists the motion of the particle, stock price

movements are driven by a growth rate, 𝜐, similar to the interest earned from investment.. Also

while the stochastic “force” randomly fluctuates the velocity of the particle, a stochastic term,

𝑑𝑊, randomly fluctuates the stock’s price. 𝑑𝑊 will be discussed further in section 4.1. These

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changes transform equation (2) into the difference equation utilized to perform Monte-Carlo

simulations for stock prices as follows:

𝑑𝑆 = 𝑆(𝑡) (𝜐 𝑑𝑡 + 𝜎 𝑑𝑊). (3)

where 𝜎 is the volatility which weights 𝑑𝑊 . Equation (3) will be discretized and utilized to

perform Monte-Carlo simulations in the next section.


4. NUMERICAL SOLUTIONS

Typically in physics it is necessary to solve differential equations analytically. A

common example is harmonic oscillations. Other examples are population growth and

continuously compounded interest. Consider the following simple ordinary differential equation:

𝑑𝑦

𝑑𝑡= 𝑟 𝑦(𝑡), (4)

In other words, the rate of growth is directly proportional to a growth rate, r. Equation (4) has a

simple exponential solution as follows,

𝑦(𝑡) = 𝑦𝑜𝑒𝑟 𝑡,

where 𝑦𝑜 is the initial value of 𝑦. The exponential nature of the example provides an excellent

starting place for finance due to the time value of money. The time value of money refers to the

cost of holding money over time. By holding onto money, the holder forgoes interest

opportunities from investing that money. The minimum interest forgone is equal to the amount

of interest generated from continuous compound interest at the risk-free interest rate.

Though we can solve an ordinary differential equation, typically it requires unnecessary

brute force to solve more difficult differential equations analytically. Many equations cannot be

solved analytically at all. Instead we can reach numerical solutions through computation by

discretizing the equation. The Taylor Series expansion of y about 𝑡 = 𝑡𝑜 is as follows:

𝑦(𝑡) = 𝑦(𝑡𝑜) +𝜕𝑦

𝜕𝑡|𝑡=𝑡𝑜

(𝑡 − 𝑡𝑜) + ⋯

𝑦𝑛+1 = 𝑦𝑛 + 𝑟 𝑦𝑛 ∆𝑡, (5)


Equation (5) can then be simplified as:

i.e. 𝑦𝑛+1 = 𝑦𝑛(1 + 𝑟 ∆𝑡). (6)

where 𝑦𝑛+1 is the output for the input 𝑦𝑛. Since equation (6) is a Taylor Series approximation

with only the first couple of terms, the time steps, ∆𝑡, must be sufficiently small for the proper

solution. With sufficiently small time steps, numeric solutions can be found based off the initial

value, 𝑦0. Figure 1 illustrates the numerical solutions that result from a numerical simulation

with equation (6).

Figure 1: Smoothed plot of the numerical solution for equation (4) with a rate of 𝑟 = 0.1 and an

initial value 𝑦𝑜 = 100. See Appendix A-2 for additional information.

More generally, equation (4) can be rewritten as 𝑑𝑦

𝑑𝑡= 𝜈(𝑡𝑛, 𝑦𝑛), then 𝑑𝑦 = 𝜈(𝑡𝑛, 𝑦𝑛)𝑑𝑡.

Discretizing this equation gives 𝑦𝑛+1 = 𝑦𝑛 + 𝜈(𝑡𝑛, 𝑦𝑛)𝑑𝑡.


4-1 RANDOM WALK

While stock prices should abide by an exponential change through time due to the time

value of money, they are also subject to many random fluctuations. These random fluctuations

can be characterized as Brownian motion, and can be taken into account with the additional

stochastic term, 𝑑𝑊:

𝑦𝑛+1 = 𝑦𝑛 + 𝜈(𝑡𝑛, 𝑦𝑛)𝑑𝑡 + 𝜎(𝑡𝑛, 𝑦𝑛)𝑑𝑊 (7)

where 𝑑𝑊 is the random process characterized as Brownian. 𝑑𝑊 is simply a random draw from

a normal distribution centered at zero with a standard deviation of √𝑑𝑡. This standard deviation

characterizes Brownian motion and will be discussed further in section 4-1. Equation (7) is a

form of general stochastic differential equation. A stochastic differential equation is simply a

differential equation which has stochastic terms (𝑑𝑊).

Consider a simple example where only the simplest of “randomness” (a volatility of 1) is

present,

𝜐(𝑡𝑛, 𝑦𝑛) = 0, 𝜎(𝑡𝑛, 𝑦𝑛) = 1, 𝑦𝑜 = 0 → 𝑦𝑛+1 = 𝑦𝑛 + 𝑑𝑊, (8)

or more simply, 𝑑𝑦 = 𝑑𝑊. This can be solved analytically as 𝑦(𝑡) = 𝑊(𝑡) + 𝑦𝑜, where 𝑦𝑜 is the

initial value of 𝑦. Since 𝑓(𝑡𝑛, 𝑦𝑛) = 0, equation (8) is strictly stochastic (nondeterministic). This

process is actually a discretized one-dimensional Brownian motion, or more simply, a one-

dimensional random walk which is centered at 𝑦𝑜 = 0 . At each time step, the particle randomly

decides whether to step a random distance, 𝑑𝑊, or remain at rest. Figure (2) is an example of a

two-dimensional random walk which may only step finite distances.


Figure 2: A visualization of a two-dimensional random walk Monte Carlo simulation made by

Erin Brown and me. At each time step, the particle may move one of four directions (up, down,

left, or right) or choose not to move. The particle begins at “*” and finishes at “&”. The trial took

300 steps and had a probability to move of 0.5.


4-2 RELATIONSHIP WITH TIME

At each step, a mean squared displacement can be calculated and then plotted as in figure

(3). A simulation can be repeated for a large number of particles to find an averaged mean

squared displacement for a given number of steps.

Figure 3: A graph of the data output from two-dimensional random walk simulations I

made, which calculates and averages the mean squared displacement at each step. The lower

curve allowed movement in four directions (separated by 90 degrees), while the upper curve

allowed movement in six directions (separated by 60 degrees). The mean squared displacement

values at each step are averaged over 1000 trials.


The mean squared displacement’s relationship with time can be found by applying the

following solution from the one-dimensional diffusion equation:

𝑃(𝑟, 𝑡) =1

√4𝜋𝐷𝑡𝑒

−(𝑟−𝑟𝑜)2

4𝐷𝑡 ,

where 𝐷 is the diffusivity constant and 𝑃 is the probability of the particle at posision 𝑟 and time

𝑡. This is simply a normal distribution centered at 𝑟𝑜 with a standard deviation of √2𝐷𝑡. Einstein

utilized this solution to derive the mean squared displacement’s relationship with time [9]:

< 𝑟2(𝑡) > = 2𝑑𝐷𝑡 (9)

where 𝑑 = 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠. Similarly in order for the linear relationship to hold

true, 𝑑𝑊 must be a random draw from a normal distribution with a standard deviation of √𝑑𝑡.

From the slopes of the best fit lines provided from figure (3) and equation (9), 𝐷 = 0.04575 for

the four-directional case and 𝐷 = 0.055145 for the six directional case. Since the mean squared

displacement is directly proportional to the number of dimensions, it intuitively makes sense that

the six directional case has a steeper slope than the four directional case.

The numerical solutions of equation (8) can be calculated with the random samplings

from 𝑑𝑊. Figure 4 illustrates the possible solutions that result from a Monte-Carlo simulation

with equation (8).


Figure 4: 50 random walks, or trials, from 0 to 20 with 2000 steps. The walks are closely related

to the plotted 𝑚√𝑡 functions as expected for a random walk where 𝑚 represents the integers

from −3 to 3. See Appendix A-3 for additional information.

Figure (4) emphasizes that the displacement is proportional to √𝑡. This is due to the fact that

𝑦(𝑡) = 𝑊(𝑡) = 𝑥√𝑡 (where 𝑥 is random draw from a normal distribution centered at 0 with a

standard deviation of 1)which comes from 𝑑𝑊 = 𝑥 √𝑑𝑡 as follows:

𝑦(𝑡) = 𝑊(𝑡) = ∑ 𝑑𝑊 = ∑(𝑥 √𝑑𝑡) = √𝑑𝑡 ∑ 𝑥. (10)

Then by applying the central limit theorem (see Appendix A), equation (10) becomes the

following:

𝑦(𝑡) = √𝑑𝑡 √𝑛 𝑥 = 𝑥√𝑛 𝑑𝑡 = 𝑥 √𝑡. (11)

This information will be crucial for analytically solving more difficult stochastic differential

equations in section 5.


4-3 STOCK PRICE EVOLUTION

The previous examples demonstrate the process of numerically solving basic differential

equations. However, the examples either relate the output purely to time (deterministic) or purely

to randomness (stochastic). Stock prices move in a way which includes both the time and

random elements. The discretized SDE from equation (3) is given by:

𝑑𝑆 = 𝑆(𝑡) (𝜐 𝑑𝑡 + 𝜎 𝑑𝑊) (3)

𝑆𝑛+1 = 𝑆𝑛 + 𝑆𝑛{𝜐 𝑑𝑡 + 𝜎 𝑑𝑊}. (12)

where 𝜐 is the stock’s growth rate and 𝜎 is the stock’s volatility. Clearly the output relates to

both time (through dt) and Brownian motion (through dW). The solution also maintains the

deterministic exponential nature while including the stochastic random Brownian fluctuations.

Figure 5 illustrates a trial that results from a Monte-Carlo simulation of equation (12).

Figure 5: A plot of stock price from time 𝑡 = 0 𝑡𝑜 20 with 2000 steps, 𝜐 = 0.1, 𝜎 = 0.1, and an

initial value of 100. The plot provides the look of what one envisions on the computer screens at

Wall Street. The plot also illustrates the exponential foundation for the stock price evolution.

This plot will be recreated to show the exponential foundation in the following section.


5. ANALYTICAL SOLUTIONS

𝑑𝑆 = 𝑆(𝑡) (𝜐 𝑑𝑡 + 𝜎 𝑑𝑊). (3)

Equation (3) can be solved analytically by making a transformation. Equation (3) shows that

the derivative of the stock price should be proportional to the current price. Hence, the solution

has an exponential relationship. The solution has an exponential foundation and fluctuates

stochastically around the exponential curve. As a result, the natural log of the stock should be

considered is the following:

𝑧 = ln(𝑆). (13)

To find the new difference equation, 𝑑𝑧, simply apply Ito’s Lemma (Appendix C) as follows:

𝑑𝑧 = (𝜕𝑧

𝜕𝑡+ 𝜇 𝑆

𝜕𝑧

𝜕𝑆+

1

2𝜎2𝑆2

𝜕2𝑧

𝜕𝑆2) 𝑑𝑡 + 𝜎 𝑆

𝜕𝑧

𝜕𝑆𝑑𝑊. (14)

Then since

𝜕𝑧

𝜕𝑆=

1

𝑆,

𝜕2𝑧

𝜕𝑆2= −

1

𝑆2,

𝜕𝑧

𝜕𝑡=

𝜕𝑧

𝜕𝑆

𝜕𝑆

𝜕𝑡=

1

𝑆

𝜕𝑆

𝜕𝑡≈ 0,

equation (14) simplifies to the following:

𝑑𝑧 = (𝜐 −1

2𝜎2) 𝑑𝑡 + 𝜎 𝑑𝑊

𝑑𝑧 = 𝜇 𝑑𝑡 + 𝜎 𝑑𝑊, (15)

where 𝜇 = 𝜐 −1

2𝜎2. Equation (15) can be solved analytically through integration as

∫ 𝑑𝑧𝑡

0= ∫ (𝜇 𝑑𝑡 + 𝜎𝑑𝑊)

𝑡

0

log[𝑆(𝑡)] = 𝜇 𝑡 + 𝜎𝑊(𝑡) + log [𝑆(0)]

𝑆(𝑡) = 𝑆(0)𝑒𝜇𝑡+𝜎𝑊(𝑡).

Finally, since 𝑊(𝑡) = 𝑥√𝑑𝑡 from section 4.1,


𝑆(𝑡) = 𝑆(0)𝑒𝜇 𝑡+𝑥 𝜎√𝑡 . (16)

Figure (6) illustrates the solution from equation (16) when plotted with 50 trials that results from

a Monte-Carlo simulation of equation (12).

Figure 6: 50 trials of the plot from figure (5) from time 𝑡 = 0 𝑡𝑜 2 with 2000 steps, 𝜐 = 0.1, 𝜎 =

0.1, and an initial value of 100. The trials are closely related to the plotted 𝑆(0)𝑒𝜇 𝑡+𝑚 𝜎√𝑡

functions as expected for a random walk where 𝑚 represents the integers from −3 to 3. See

Appendix B-4 for additional information.

5-1 BLACK-SCHOLES EQUATION

While Monte Carlo simulation provides a solution, another simpler solution is available

for less complex options. Consider the value of an option, V(t,S), which is related to the stock

price, S.

The Black-Scholes equation is based off a delta-hedging portfolio, X. A portfolio is simply a

collection of financial assets or investments. The portfolio for Black-Scholes is just a collection


of stocks and option contracts. The portfolio, X, is either short one option contract and long 𝜕𝑉

𝜕𝑆

stocks as

𝑋 = −𝑉 +𝜕𝑉

𝜕𝑆𝑆, (17)

or long one option and short 𝜕𝑉

𝜕𝑆 stocks as

𝑋 = 𝑉 −𝜕𝑉

𝜕𝑆𝑆, (18)

where 𝜕𝑉

𝜕𝑆 is delta which measures the sensitivity of the option value to changes in the underlying

stock’s value assuming all else constant. As a result of delta-hedging, the overall portfolio’s

value will not change from small changes in the stock price. Essentially risk is nullified by

offsetting the option position by owning stock based on the option’s sensitivity to stock price

changes. To derive the Black-Scholes equation, discretize equation (18):

𝛥𝑋 = −𝛥𝑉 +𝜕𝑉

𝜕𝑆𝛥𝑆. (20)

Since 𝑑𝑆 ≈ 𝛥𝑆, equation (3) can be discretized as follows:

𝑑𝑆 = 𝑆(𝑡) (𝜐 𝑑𝑡 + 𝜎 𝑑𝑊)

∆𝑆 = 𝑆[𝜇 ∆𝑡 + 𝜎 ∆𝑊]. (21)

Since 𝑑𝑉 ≈ 𝛥𝑉, 𝛥𝑉 can be found by applying the identity Ito’s Lemma (Appendix A) for 𝑑𝑉 as

follows:

𝑑𝑉 = (𝜕𝑉

𝜕𝑡+ 𝜇 𝑆

𝜕𝑉

𝜕𝑆+

1

2𝜎2 𝑆2 𝜕2𝑉

𝜕𝑆2) 𝑑𝑡 + 𝜎 𝑆𝜕𝑉

𝜕𝑆𝑑𝑊

∆𝑉 = (𝜕𝑉

𝜕𝑡+ 𝜇 𝑆

𝜕𝑉

𝜕𝑆+

1

2𝜎2 𝑆2 𝜕2𝑉

𝜕𝑆2) ∆𝑡 + 𝜎 𝑆𝜕𝑉

𝜕𝑆∆𝑊 (22)

Then after substituting equations (21) and (22), equation (20) becomes

𝛥𝑋 = (−𝜕𝑉

𝜕𝑡−

1

2𝜎2 𝑆2 𝜕2𝑉

𝜕𝑆2) 𝛥𝑡 (23)


Notice that the specific choice of portfolio allowed for the “random” factor of dW to be averted

in equation (23). This is what causes delta-hedged portfolios to reduce risk. However, risk is not

eliminated because 𝛥𝑋 includes S(t) which is a random process. Since the risk is severely

reduced, the return on the portfolio, 𝛥𝑋, is simply the simple compound interest generated at the

risk-free interest rate, r. and so

𝛥𝑋 = 𝑋 𝑟 𝛥𝑡 = 𝑟 (−𝑉 +𝜕𝑉

𝜕𝑆𝑆) 𝛥𝑡. (24)

Simple compound interest is similar to calculating displacement from velocity (m/s) and time

interval (s) in physics. The velocity is analogous to the principle ($) times the rate (%/yr).

Combining equations (23) and (24) results in the general Black-Scholes equation,

𝜕𝑉

𝜕𝑡+ 𝑟𝑆

𝜕𝑉

𝜕𝑆+

1

2𝜎2𝑆2 𝜕2𝑉

𝜕𝑆2− 𝑟𝑉 = 0. (25)

5-2 BLACK-SCHOLES FORMULA

Solving the Black-Scholes equation requires some degree of brute force through

transformations. However, this process is important because it reaches an easier to solve

diffusion equation which thermal physicists are familiar. In order to change the coefficients of

the terms in equation (25) into constants, there must be a change of variable to x=ln(S/K) where

𝜕𝑉

𝜕𝑆=

𝜕𝑉

𝜕𝑥

1

𝑆 and

𝜕2𝑉

𝜕𝑆2=

1

𝑆2(

𝜕2𝑉

𝜕𝑥2−

𝜕𝑉

𝜕𝑥).

Since the value of the option at maturity is known, there must also be a change of variable from t

to T-t. As a result of these changes, equation (25) becomes

−𝜕𝑉

𝜕(𝑇−𝑡)+ (𝑟 −

1

2𝜎2)

𝜕𝑉

𝜕𝑥+

1

2𝜎2 𝜕2𝑉

𝜕𝑥2− 𝑟𝑉 = 0, (26)

Next, the final term can be absorbed with the transformation

𝑈 = 𝑉 𝑒𝑟(𝑇−𝑡).


Now equation (26) becomes

𝑒−𝑟(𝑇−𝑡) [−𝜕𝑈

𝜕(𝑇−𝑡)+ 𝑟𝑈 + (𝑟 −

1

2𝜎2)

𝜕𝑈

𝜕𝑥+

1

2𝜎2 𝜕2𝑈

𝜕𝑥2] − 𝑟𝑉 = 0,

which simplifies to

−𝜕𝑈

𝜕(𝑇−𝑡)+ (𝑟 −

1

2𝜎2)

𝜕𝑈

𝜕𝑥+

1

2𝜎2 𝜕2𝑈

𝜕𝑥2= 0. (27)

In an effort to remove the coefficients, the following transformation must be done:

𝑦 =𝑟−𝜎2/2

𝜎2/2𝑥.

This alters equation (27) into

−𝜕𝑈

𝜕(𝑇−𝑡)+ (𝑟 −

1

2𝜎2) [

𝜕𝑈

𝜕𝑦

𝑟−𝜎2

2𝜎2

2

] +1

2𝜎2 [

𝜕𝑈

𝜕𝑦

𝑟−𝜎2

2𝜎2

2

]

2

= 0

and simplifies to

−𝜕𝑈

𝜕𝜏+

𝜕𝑈

𝜕𝑦+

𝜕2𝑈

𝜕𝑦2= 0 (28)

where

𝜏 =(𝑟−𝜎2/2)

2

𝜎2/2(𝑇 − 𝑡). (29)

The final transformation is the change of variable 𝑧 = 𝜏 + 𝑦. Since 𝑈(𝜏, 𝑧(𝜏, 𝑦)) rather than just

𝑈(𝜏, 𝑦), the terms from equation (28) become:

𝜕𝑈(𝜏,𝑦)

𝜕𝜏=

𝜕𝑈(𝜏,𝑧(𝜏,𝑦))

𝜕𝜏+


𝜕𝑧

𝜕𝑧

𝜕𝜏=


𝜕𝜏+


𝜕𝑧, (30)

𝜕𝑈(𝜏,𝑦)

𝜕𝑦=


𝜕𝑧

𝜕𝑧

𝜕𝑦=


𝜕𝑧, (31)

𝜕2𝑈(𝜏,𝑦)

𝜕𝑦2=

𝜕

𝜕𝑦[


𝜕𝑧] =

𝜕2𝑈(𝜏,𝑧(𝜏,𝑦))

𝜕𝑧2

𝜕𝑧

𝜕𝑦=

𝜕2𝑈(𝜏,𝑧(𝜏,𝑦))

𝜕𝑧2. (32)

Substituting equations (30), (31), and (32) into equation (28) yields the diffusion equation,


𝜕𝑈

𝜕𝜏=

𝜕2𝑈

𝜕𝑧2. (33)

Equation (33) is a basic one-dimensional diffusion equation which can be related to both the

diffusion equation for Brownian motion and a heat diffusion equation. U, z, and 𝜏 are loosely the

option value, stock price, and time respectively. As a result, equation (33) means that the option

value diffuses along the stock price through time similar to how the Brownian particle and heat

diffuse along spatial directions through time. Equation (33) can be solved and transformed back

into familiar terms as the Black-Scholes formula for European call or put options:

𝑉𝑐(𝑆, 𝑡) = 𝑆(𝑡)𝑁(𝑑1) − 𝑋𝑒−𝑟(𝑇−𝑡)𝑁(𝑑2) (34)

𝑉𝑝(𝑆, 𝑡) = 𝑋𝑒−𝑟(𝑇−𝑡)𝑁(−𝑑2) − 𝑆(𝑡)𝑁(−𝑑1) (35)

where 𝑉𝑐(𝑆, 𝑡) is the price of a call option, 𝑉𝑝(𝑆, 𝑡) is the price of a put option, X is the strike

price, T is the maturity date, N(x) is a probability density function (PDF) such that

𝑁(𝑥) = ∫1

√2𝜋𝑒

−𝑦2

2⁄ 𝑑𝑦

∞

−𝑥 (36)

and

𝑑1 =ln(𝑆(0)

𝑋⁄ )+(𝑟+𝜎2

2⁄ )𝑇

𝜎√𝑇 (37) 𝑑2 =

ln(𝑆(0)𝑋⁄ )+(𝑟−𝜎2

2⁄ )𝑇

𝜎√𝑇. (38)

5-3 UNDERSTANDING THE BLACK-SCHOLES SOLUTION

While equations (34) and (35) may seem daunting, they are actually fairly intuitive.

Equation (34) essentially states that the current value of the call option, 𝑉𝑐, is the current price of

the stock, 𝑆(𝑡), minus the exercise price of the stock, 𝑋, discounted back to current value. The

discounting of the exercise price back to current value results from the theoretical cost of holding

money which comes from forgone interest opportunities as mentioned in section 4. This

theoretical cost is handled through continuous compounding (𝑒−𝑟(𝑇−𝑡)) at the risk-free interest


rate, r. Equation (35) is basically the opposite of equation (34) where the current value of the call

option is the exercise price of the stock discounted back to current value minus the current price

of the stock. This makes sense since a call option is the right to buy at the exercise price and then

sell at the current stock price while a put option is the right to sell at the exercise price after

buying at the current stock price. Both the current price and exercise price are weighted by

similar respective PDFs, N. In laymen’s terms, the functions can be thought as the probabilities

of executing the option.

There are a few parameters in 𝑑1 and 𝑑2 to evaluate. First is the ratio of the initial stock

price with the strike price, 𝑆(0) 𝑋⁄ . Both 𝑑1 and 𝑑2 increase by the same amount with respect to

an increase in the ratio. As a result, the PDFs increase by the same amount for equation (34) and

decrease by the same amount for equation (35). This in turn increases the value of the call and

decreases the value of the put. This makes sense because the goal of the call owner is for the

stock price to increase above the exercise price and the goal of the put owner is for the stock

price to decrease below the exercise price. A larger ratio benefits the call owner’s cause and

hinders the put owner’s cause.

The risk-free interest rate, r, and the maturity date, T, have a similar effect on the option

value as the ratio of the initial stock price with the strike price. When r or T increase, the value of

the call decreases and the value of the put decreases. This effect primarily occurs from 𝑑1 and

𝑑2.

It also makes sense that the option value increases as the volatility, 𝜎, increases. Since the

volatility represents the significance of ‘noise’ or fluctuations in the price, a larger volatility is

good for the price of the option because a large fluctuation can occur at maturity in favor of the


trader causing a huge gain. If the fluctuation does not occur in favor of the trader, then there is

only a small loss from the premium.


6. FUTURE WORK

While the Black-Scholes solution provides a quick and simple value for the option, it is

not necessarily preferred over Monte Carlo methods. Monte Carlo methods with a large enough

number of simulations can provide a distribution of values from all of the simulations. This

distribution can not only provide the same value found from the Black-Scholes solution, but also

how much different the observed value can be and the likelihood of such a difference. Many

current models, such as the Black-Scholes solution, assume that certain variables are constant

throughout the lifetime of the option. However, in reality variables like volatility can vary during

the option contract’s lifetime.

The future of this project aims to explore the capabilities of Monte Carlo methods beyond

that of section 4, specifically the effects of slight variation in the volatility. The project can

compare several different Monte Carlo pricing methods which only differ on the degree to which

they fluctuate the volatility. To fluctuate the volatility, the simulation can simply perform a

random draw for the volatility at each time step similar to dW. The standard deviation of these

random draws can be a user input percentage of the volatility. The stock price can then be

trajected similar to figure (5). Once the final stock value is averaged for enough trials, the option

value is simply the final stock price (discounted back to current value) minus the initial stock

price. While creating this simulation is straightforward, further exploration needs to be done on

valuing the drift, 𝜐, and volatility, 𝜎. I found that increasing the number of time steps caused the

option price to approach zero. I believe this was because it over emphasized the stochastic

contributions and for a large amount of trials the option value averaged zero (the expectation

value resulting from the random draws of dW).


The pricing methods should be compared against the Black-Scholes pricing model to

determine which method is the most accurate. The models can be tested on hypothetical option

contracts based off of historical stock prices. These hypothetical option contracts can be made

from the historical daily stock prices from Yahoo Finance. After a maturity date is chosen, each

day can provide a theoretical option price based on the difference between the current stock price

and the stock price at the chosen maturity date (discounted back to current date). These

theoretical option prices can be utilized to test the various pricing models.


7. APPENDIX

A. MATHEMATICA

A-1 GLOSSARY

Evaluate[expr]

Causes expr to be evaluated even if it appears as the argument of a function whose

attributes specify that it should be held unevaluated.

ItoProcess[sdeqns, expr, x, t, 𝒘 ≈dproc]

Represents an Ito process specified by a stochastic differential equation sdeqns, output

expression expr, with state x and time t, driven by w following the process dproc.

ListLinePlot[{𝒍𝒊𝒔𝒕𝟏, 𝒍𝒊𝒔𝒕𝟐, … }]

Plots several lines.

ListPlot[{𝒚𝟏, 𝒚𝟐, … }]

Plots points corresponding to a list of values, assumed to correspond to x coordinates 1, 2,

… .

NDSolve[eqns, n, {x, 𝒙𝒎𝒊𝒏, 𝒙𝒎𝒂𝒙}]

Finds a numerical solution to the ordinary differential equation eqns for the function u

with the independent variable x in the range 𝒙𝒎𝒊𝒏 to 𝒙𝒎𝒂𝒙.


NestList[f, expr, n]

Gives a list of the results of applying f to expr 0 through n times.

Plot[{𝒇𝟏, 𝒇𝟐, …}, { x, 𝒙𝒎𝒊𝒏, 𝒙𝒎𝒂𝒙}]

Plots several functions 𝑓𝑖.

RandomFunction[proc, {𝒕𝒎𝒊𝒏, 𝒕𝒎𝒂𝒙}]

Generates a pseudorandom function from the process proc from 𝑡𝑚𝑖𝑛 to 𝑡𝑚𝑎𝑥.

Show[𝒈𝟏, 𝒈𝟐, …]

Shows several graphics combined.

Table[expr, {𝒊𝒎𝒂𝒙}]

Generates a list of 𝑖𝑚𝑎𝑥 copies of expr.

A-2 FIGURE (1)


A-3 FIGURE (4)

A-4 FIGURE (6)

B. CENTRAL LIMIT THEOREM

Let {𝑞𝑗}𝑗𝜖𝑁 be a sequence of independent, identically distributed random variables

random variables, let 𝜂 be the expectation value for 𝑥𝑗, let 𝜎 be the standard deviation for 𝑥𝑗, and

define

𝑄𝑛 = ∑ 𝑞𝑗

𝑛

𝑗=1

.


The central limit theorem states that as 𝑛 → ∞ [10],

𝑄𝑛 − 𝑛 𝜂

√𝑛 𝜎2→ 𝑥,

where x is a draw from a standard normal distribution (mean of 0 and standard deviation of 1).

The central limit theorem can also be rewritten as

𝑄𝑛 ≈ 𝑥√𝑛 𝜎2 + 𝑛 𝜂.

If 𝑞𝑗 = 𝑥, then 𝜂 = 0 and 𝜎 = 1. As a result, the central limit theorem can be simplified as

𝑄𝑛 = ∑ 𝑥 ≈ 𝑥√𝑛 .

C. ITO’S LEMMA

Ito’s Lemma is the stochastic calculus equivalent of the chain rule from calculus. It

serves as a tool to find the differential of a stochastic process. Given the function, f(t,x), the

Taylor series expansion is:

𝑑𝑓 =𝜕𝑓

𝜕𝑡𝑑𝑡 +

𝜕𝑓

𝜕𝑥𝑑𝑥 +

1

2

𝜕2𝑓

𝜕𝑥2𝑑𝑥2 + ⋯

where dx is the stochastic difference equation,

𝑑𝑥 = 𝑥(𝜐 𝑑𝑡 + 𝜎 𝑑𝑊).

Substituting in for dx results in

𝑑𝑓 =𝜕𝑓

𝜕𝑡𝑑𝑡 +

𝜕𝑓

𝜕𝑥 𝑥 (𝜐 𝑑𝑡 + 𝜎 𝑑𝑊) +

1

2

𝜕2𝑓

𝜕𝑥2 𝑥2 (𝜐2 𝑑𝑡2 + 2𝜐 𝜎 𝑑𝑡 𝑑𝑊 + 𝜎2 𝑑𝑊2).

Since 𝑑𝑡2 ≈ 0, 𝑑𝑡 𝑑𝑊 ≈ 0, and 𝑑𝑊2 ≈ 𝑑𝑡, this simplifies to

𝑑𝑓 = (𝜕𝑓

𝜕𝑡 + 𝜐 𝑥

𝜕𝑓

𝜕𝑥 +

𝜎2

2 𝑥2

𝜕2𝑓

𝜕𝑥2) 𝑑𝑡 + 𝜎 𝑥

𝜕𝑓

𝜕𝑥 𝑑𝑊.


C-1 𝒅𝑾𝟐 ≈ 𝒅𝒕

From the definition of variance,

𝑉(𝑑𝑊) = 𝐸(𝑑𝑊2) − 𝐸(𝑑𝑊)2

where 𝑉(𝑑𝑊) is the variance of dW and 𝐸(𝑑𝑊) is the expectation value of dW. Then since

𝐸(𝑑𝑊) and 𝑉(𝑑𝑊) = 𝑑𝑡,

𝐸(𝑑𝑊2) = 𝑉(𝑑𝑊) = 𝑑𝑡.

Then as 𝑑𝑡 → 0, the standard deviation of 𝑑𝑊2 also decreases and hence the normal distribution

from which it draws narrows in on the expectation value, 𝑑𝑡. Thus for sufficiently small dt,

𝑑𝑊2 ≈ 𝑑𝑡.


D. WORKS CITED

[1] Gora, “The Theory of Brownian Motion: A Hundred Years’ Anniversary”, (2006),

52-53.

[2] Ibid.

[3] David Forfar, “Louis Bachelier”, (JOC/EFR, August 2002), http://www-

history.mcs.st-and.ac.uk/Biographies/Bachelier.html.

[4] Ibid.

[5] Capinski and Zastawniak, “Mathematics for Finance: An Introduction to Financial

Engineering”, Options: General Properties, (USA: Springer, 2003), 147-149.



[7] Jarrow and Protter, “A short history of stochastic integration and mathematical

finance the early years, 1880-1970”, (2004).



[9] Ebeling, “Nonlinear Brownian motion- mean square displacement”, (2004).

[10] Durrett, “Stochastic Calculus: A Practical Introduction”, (CRC Press, 1996).


E. BIBLIOGRAPHY

Capinski and Zastawniak. Mathematics for Finance: An Introduction to

Financial Engineering. USA: Springer, 2003.

Copped. “Solving the Black-Scholes equation: a demystification.” Last modified November,

2009. http://www.francoiscoppex.com/blackscholes.pdf.

Cotterill, Rodney. "Transport Process." In Biophysics an Introduction, 76-79. Chichester, West

Sussex: John Wiley, 2002.

"Forward Contract vs. Futures Contract." Diffen. Accessed September 9, 2014.

http://www.diffen.com/difference/Forward_Contract_vs_Futures_Contract

Hull. Options, Futures, and Other Derivatives. USA: Pearson Prentice Hall , 2009.

“Introduction to the Black-Scholes formula.” Khan Academy. Accessed October, 2014.

https://www.khanacademy.org/economics-finance-domain/core-finance/derivative-

securities/Black-Scholes/v/introduction-to-the-black-scholes-formula

Stojanovic. Computational Financial Mathematics. USA: Springer Science, 2003.

Application of Monte Carlo Methods in Finance

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